Lagrangian descriptors with uncertainty. (English) Zbl 07893209
Summary: Lagrangian descriptors provide a global dynamical picture of the geometric structures for arbitrarily time-dependent flows with broad applications. This paper develops a mathematical framework for computing Lagrangian descriptors when uncertainty appears. The uncertainty originates from estimating the underlying flow field as a natural consequence of data assimilation or statistical forecast. It also appears in the resulting Lagrangian trajectories. The uncertainty in the flow field directly affects the path integration of the crucial nonlinear positive scalar function in computing the Lagrangian descriptor, making it fundamentally different from many other diagnostic methods. Despite being highly nonlinear and non-Gaussian, closed analytic formulae are developed to efficiently compute the expectation of such a scalar function due to the uncertain velocity field by exploiting suitable approximations. A rapid and accurate sampling algorithm is then built to assist the forecast of the probability density function (PDF) of the Lagrangian trajectories. Such a PDF provides the weight to combine the Lagrangian descriptors along different paths. Simple but illustrative examples are designed to show the distinguished behavior of using Lagrangian descriptors in revealing the flow field when uncertainty appears. Uncertainty can either completely erode the flow3 structure or barely affect the underlying geometry of the flow field. The method is also applied for eddy identification, indicating that uncertainty has distinct impacts on detecting eddies at different time scales. Finally, when uncertainty is incorporated into the Lagrangian descriptor for inferring the source target, the likelihood criterion provides a very different conclusion from the deterministic methods.
MSC:
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
| 76F65 | Direct numerical and large eddy simulation of turbulence |
Keywords:
uncertainty; turbulent dynamical systems; Lagrangian data assimilation; eddy detection; inference of source targetSoftware:
EnKFReferences:
| [1] | Wiggins, S., Normally Hyperbolic Invariant Manifolds in Dynamical Systems, 1994, Springer Science & Business Media · Zbl 0812.58001 |
| [2] | Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics, 2017, Cambridge University Press · Zbl 1374.86002 |
| [3] | Strogatz, S. H., Nonlinear Dynamics and Chaos with Student Solutions Manual: with Applications to Physics, Biology, Chemistry, and Engineering, 2018, CRC Press |
| [4] | Wilcox, D. C., Multiscale model for turbulent flows, AIAA J., 26, 11, 1311-1320, 1988 |
| [5] | Sheard, S. A.; Mostashari, A., Principles of complex systems for systems engineering, Syst. Eng., 12, 4, 295-311, 2009 |
| [6] | Ghil, M.; Childress, S., Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, 2012, Springer Science & Business Media · Zbl 0643.76001 |
| [7] | Farazmand, M.; Sapsis, T. P., Extreme events: Mechanisms and prediction, Appl. Mech. Rev., 71, 5, 2019 |
| [8] | Trenberth, K. E.; Fasullo, J. T.; Shepherd, T. G., Attribution of climate extreme events, Nature Clim. Change, 5, 8, 725-730, 2015 |
| [9] | Moffatt, H., Extreme events in turbulent flow, J. Fluid Mech., 914, F1, 2021 · Zbl 1461.76183 |
| [10] | Majda, A., Introduction to PDEs and Waves for the Atmosphere and Ocean, 2003, American Mathematical Soc. · Zbl 1278.76004 |
| [11] | Manneville, P.; Pomeau, Y., Intermittency and the Lorenz model, Phys. Lett. A, 75, 1-2, 1-2, 1979 · Zbl 0985.37503 |
| [12] | Asch, M.; Bocquet, M.; Nodet, M., Data Assimilation: Methods, Algorithms, and Applications, 2016, SIAM · Zbl 1361.93001 |
| [13] | Kalnay, E., Atmospheric Modeling, Data Assimilation and Predictability, 2003, Cambridge University Press |
| [14] | Majda, A. J.; Harlim, J., Filtering Complex Turbulent Systems, 2012, Cambridge University Press · Zbl 1250.93002 |
| [15] | Law, K.; Stuart, A.; Zygalakis, K., Data Assimilation, 2015, Springer: Springer Cham, Switzerland · Zbl 1353.60002 |
| [16] | Ghil, M.; Malanotte-Rizzoli, P., Data assimilation in meteorology and oceanography, (Advances in Geophysics, vol. 33, 1991, Elsevier), 141-266 |
| [17] | Mendoza, C.; Mancho, A. M., Hidden geometry of ocean flows, Phys. Rev. Lett., 105, 3, Article 038501 pp., 2010 |
| [18] | Madrid, J. J.; Mancho, A. M., Distinguished trajectories in time dependent vector fields, Chaos, 19, 1, Article 013111 pp., 2009 |
| [19] | Lopesino, C.; Balibrea-Iniesta, F.; García-Garrido, V. J.; Wiggins, S.; Mancho, A. M., A theoretical framework for Lagrangian descriptors, Int. J. Bifurcation Chaos, 27, 01, Article 1730001 pp., 2017 · Zbl 1358.34048 |
| [20] | Mancho, A. M.; Wiggins, S.; Curbelo, J.; Mendoza, C., Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18, 12, 3530-3557, 2013 · Zbl 1344.37031 |
| [21] | Demian, A. S.; Wiggins, S., Detection of periodic orbits in Hamiltonian systems using Lagrangian descriptors, Int. J. Bifurcation Chaos, 27, 14, Article 1750225 pp., 2017 · Zbl 1386.37061 |
| [22] | Naik, S.; García-Garrido, V. J.; Wiggins, S., Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors, Commun. Nonlinear Sci. Numer. Simul., 79, Article 104907 pp., 2019 · Zbl 1522.37088 |
| [23] | García-Garrido, V. J.; Naik, S.; Wiggins, S., Tilting and squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics, Int. J. Bifurcation Chaos, 30, 04, Article 2030008 pp., 2020 · Zbl 1442.37069 |
| [24] | Wiggins, S.; Wiggins, S.; Golubitsky, M., Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2003, Springer · Zbl 1027.37002 |
| [25] | Shadden, S. C.; Lekien, F.; Marsden, J. E., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 3-4, 271-304, 2005 · Zbl 1161.76487 |
| [26] | Haller, G., A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240, 7, 574-598, 2011 · Zbl 1214.37056 |
| [27] | Haller, G.; Hadjighasem, A.; Farazmand, M.; Huhn, F., Defining coherent vortices objectively from the vorticity, J. Fluid Mech., 795, 136-173, 2016 · Zbl 1359.76096 |
| [28] | Balasuriya, S., Stochastic sensitivity: A computable Lagrangian uncertainty measure for unsteady flows, SIAM Rev., 62, 4, 781-816, 2020 · Zbl 1459.76108 |
| [29] | Froyland, G.; Santitissadeekorn, N.; Monahan, A., Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20, 4, 2010 · Zbl 1311.37008 |
| [30] | Froyland, G., An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems, Physica D, 250, 1-19, 2013 · Zbl 1355.37013 |
| [31] | Bezdek, J. C.; Ehrlich, R.; Full, W., FCM: The fuzzy c-means clustering algorithm, Comput. Geosci., 10, 2-3, 191-203, 1984 |
| [32] | Bezdek, J. C.; Hathaway, R. J.; Sabin, M. J.; Tucker, W. T., Convergence theory for fuzzy c-means: counterexamples and repairs, IEEE Trans. Syst. Man Cybern., 17, 5, 873-877, 1987 · Zbl 0653.68091 |
| [33] | Schlueter-Kuck, K. L.; Dabiri, J. O., Coherent structure colouring: identification of coherent structures from sparse data using graph theory, J. Fluid Mech., 811, 468-486, 2017 · Zbl 1383.76372 |
| [34] | Vortmeyer-Kley, R.; Gräwe, U.; Feudel, U., Detecting and tracking eddies in oceanic flow fields: a Lagrangian descriptor based on the modulus of vorticity, Nonlinear Process. Geophys., 23, 4, 159-173, 2016 |
| [35] | Mendoza, C.; Mancho, A.; Wiggins, S., Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets, Nonlinear Process. Geophys., 21, 3, 677-689, 2014 |
| [36] | García-Sánchez, G.; Mancho, A. M.; Ramos, A. G.; Coca, J.; Wiggins, S., Structured pathways in the turbulence organizing recent oil spill events in the Eastern Mediterranean, Sci. Rep., 12, 1, 3662, 2022 |
| [37] | de La Cámara, A.; Mancho, A. M.; Ide, K.; Serrano, E.; Mechoso, C. R., Routes of transport across the antarctic polar vortex in the Southern spring, J. Atmos. Sci., 69, 2, 741-752, 2012 |
| [38] | Curbelo, J.; Mechoso, C. R.; Mancho, A. M.; Wiggins, S., Lagrangian study of the final warming in the Southern stratosphere during 2002: Part I. The vortex splitting at upper levels, Clim. Dyn., 53, 2779-2792, 2019 |
| [39] | Curbelo, J.; Mechoso, C. R.; Mancho, A. M.; Wiggins, S., Lagrangian study of the final warming in the Southern stratosphere during 2002: Part II. 3D structure, Clim. Dyn., 53, 1277-1286, 2019 |
| [40] | Craven, G. T.; Hernandez, R., Lagrangian descriptors of thermalized transition states on time-varying energy surfaces, Phys. Rev. Lett., 115, 14, Article 148301 pp., 2015 |
| [41] | Craven, G. T.; Junginger, A.; Hernandez, R., Lagrangian descriptors of driven chemical reaction manifolds, Phys. Rev. E, 96, 2, Article 022222 pp., 2017 |
| [42] | García-Garrido, V. J.; Agaoglou, M.; Wiggins, S., Exploring isomerization dynamics on a potential energy surface with an index-2 saddle using Lagrangian descriptors, Commun. Nonlinear Sci. Numer. Simul., 89, Article 105331 pp., 2020 · Zbl 1477.37090 |
| [43] | Naik, S.; Wiggins, S., Detecting reactive islands in a system-bath model of isomerization, Phys. Chem. Chem. Phys., 22, 32, 17890-17912, 2020 |
| [44] | Majda, A. J.; Branicki, M., Lessons in uncertainty quantification for turbulent dynamical systems, Discrete Contin. Dyn. Syst., 32, 9, 3133, 2012 · Zbl 1264.94076 |
| [45] | Mignolet, M. P.; Soize, C., Stochastic reduced order models for uncertain geometrically nonlinear dynamical systems, Comput. Methods Appl. Mech. Engrg., 197, 45-48, 3951-3963, 2008 · Zbl 1194.74154 |
| [46] | Majda, A. J.; Chen, N., Model error, information barriers, state estimation and prediction in complex multiscale systems, Entropy, 20, 9, 644, 2018 |
| [47] | Majda, A. J., Introduction to Turbulent Dynamical Systems in Complex Systems, 2016, Springer · Zbl 1377.37003 |
| [48] | Palmer, T. N., A nonlinear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parametrization in weather and climate prediction models, Q. J. R. Meteorol. Soc., 127, 572, 279-304, 2001 |
| [49] | Givon, D.; Kupferman, R.; Stuart, A., Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity, 17, 6, R55, 2004 · Zbl 1073.82038 |
| [50] | Trémolet, Y., Model-error estimation in 4D-Var, Q. J. R. Meteorol. Soc.: a J. Atmos. Sci., Appl. Meteorol. Phys. Oceanogr., 133, 626, 1267-1280, 2007 |
| [51] | Qiu, B.; Chen, S.; Klein, P.; Torres, H.; Wang, J.; Fu, L.-L.; Menemenlis, D., Reconstructing upper-ocean vertical velocity field from sea surface height in the presence of unbalanced motion, J. Phys. Oceanogr., 50, 1, 55-79, 2020 |
| [52] | Liu, Y.; Weisberg, R. H., Patterns of ocean current variability on the West Florida Shelf using the self-organizing map, J. Geophys. Res.: Oceans, 110, C6, 2005 |
| [53] | Doglioni, F.; Ricker, R.; Rabe, B.; Kanzow, T., Sea surface height anomaly and geostrophic velocity from altimetry measurements over the Arctic Ocean (2011-2018), Earth Syst. Sci. Data Discuss., 1-46, 2021 |
| [54] | Orrell, D.; Smith, L.; Barkmeijer, J.; Palmer, T. N., Model error in weather forecasting, Nonlinear Process. Geophys., 8, 6, 357-371, 2001 |
| [55] | Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 4, 483-531, 2015 · Zbl 1339.37089 |
| [56] | Evensen, G., Data Assimilation: the Ensemble Kalman Filter, 2009, Springer · Zbl 1395.93534 |
| [57] | Apte, A.; Jones, C., The impact of nonlinearity in Lagrangian data assimilation, Nonlinear Process. Geophys., 20, 3, 329-341, 2013 |
| [58] | Apte, A.; Jones, C. K.; Stuart, A.; Voss, J., Data assimilation: Mathematical and statistical perspectives, Int. J. Numer. Methods Fluids, 56, 8, 1033-1046, 2008 · Zbl 1384.62300 |
| [59] | Apte, A.; Jones, C. K.; Stuart, A., A Bayesian approach to Lagrangian data assimilation, Tellus A: Dyn. Meteorol. Oceanogr., 60, 2, 336-347, 2008 |
| [60] | Ide, K.; Kuznetsov, L.; Jones, C. K., Lagrangian data assimilation for point vortex systems, J. Turbul., 3, 1, 053, 2002 · Zbl 1083.76531 |
| [61] | Chen, N.; Majda, A. J.; Tong, X. T., Information barriers for noisy Lagrangian tracers in filtering random incompressible flows, Nonlinearity, 27, 9, 2133, 2014 · Zbl 1320.60102 |
| [62] | Manucharyan, G. E.; Thompson, A. F., Submesoscale sea ice-ocean interactions in marginal ice zones, J. Geophys. Res.: Oceans, 122, 12, 9455-9475, 2017 |
| [63] | Covington, J.; Chen, N.; Wilhelmus, M. M., Bridging gaps in the climate observation network: A physics-based nonlinear dynamical interpolation of Lagrangian ice floe measurements via data-driven stochastic models, J. Adv. Modelling Earth Syst., 14, 9, 2022, 2022MS003218 |
| [64] | Maclean, J.; Santitissadeekorn, N.; Jones, C. K., A coherent structure approach for parameter estimation in Lagrangian Data Assimilation, Physica D, 360, 36-45, 2017 · Zbl 1378.93132 |
| [65] | Hadjighasem, A.; Karrasch, D.; Teramoto, H.; Haller, G., Spectral-clustering approach to Lagrangian vortex detection, Phys. Rev. E, 93, 6, Article 063107 pp., 2016 |
| [66] | Froyland, G.; Padberg-Gehle, K., A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data, Chaos, 25, 8, Article 087406 pp., 2015 · Zbl 1374.37114 |
| [67] | Lahoz, B. K.W.; Menard, R., Data Assimilation, 2010, Springer · Zbl 1194.86002 |
| [68] | Badza, A.; Mattner, T. W.; Balasuriya, S., How sensitive are Lagrangian coherent structures to uncertainties in data?, Physica D, 444, Article 133580 pp., 2023 · Zbl 1503.37090 |
| [69] | Schneider, D.; Fuhrmann, J.; Reich, W.; Scheuermann, G., A variance based FTLE-like method for unsteady uncertain vector fields, (Topological Methods in Data Analysis and Visualization II: Theory, Algorithms, and Applications, 2011, Springer), 255-268 · Zbl 1426.76637 |
| [70] | Guo, H.; He, W.; Peterka, T.; Shen, H.-W.; Collis, S. M.; Helmus, J. J., Finite-time Lyapunov exponents and Lagrangian coherent structures in uncertain unsteady flows, IEEE Trans. Vis. Comput. Graph., 22, 6, 1672-1682, 2016 |
| [71] | BozorgMagham, A. E.; Ross, S. D., Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty, Commun. Nonlinear Sci. Numer. Simul., 22, 1-3, 964-979, 2015 |
| [72] | You, G.; Leung, S., Computing the finite time Lyapunov exponent for flows with uncertainties, J. Comput. Phys., 425, Article 109905 pp., 2021 · Zbl 07508501 |
| [73] | Balasuriya, S., Uncertainty in finite-time Lyapunov exponent computations, J. Comput. Dyn., 7, 2, 313-337, 2020 · Zbl 1450.37077 |
| [74] | Rapp, T.; Dachsbacher, C., Uncertain transport in unsteady flows, (2020 IEEE Visualization Conference. 2020 IEEE Visualization Conference, VIS, 2020, IEEE), 16-20 |
| [75] | García-Sánchez, G.; Mancho, A. M.; Wiggins, S., A bridge between invariant dynamical structures and uncertainty quantification, Commun. Nonlinear Sci. Numer. Simul., 104, Article 106016 pp., 2022 · Zbl 1483.37114 |
| [76] | Garcia-Sanchez, G.; Ana Maria, M.; Makrina, A.; Stephen, W., New links between invariant dynamical structures and uncertainty quantification, Physica D, 2023, Under Revision · Zbl 1516.37134 |
| [77] | Branicki, M.; Uda, K., Lagrangian uncertainty quantification and information inequalities for stochastic flows, SIAM/ASA J. Uncertain. Quantif., 9, 3, 1242-1313, 2021 · Zbl 1481.60104 |
| [78] | Branicki, M.; Uda, K., Path-based divergence rates and Lagrangian uncertainty in stochastic flows, SIAM J. Appl. Dyn. Syst., 22, 1, 419-482, 2023 · Zbl 1515.37008 |
| [79] | Chen, N.; Majda, A. J.; Tong, X. T., Noisy Lagrangian tracers for filtering random rotating compressible flows, J. Nonlinear Sci., 25, 3, 451-488, 2015 · Zbl 1329.76128 |
| [80] | Gardiner, C. W., Handbook of Stochastic Methods, 1985, springer Berlin |
| [81] | Farrell, B. F.; Ioannou, P. J., Stochastic forcing of the linearized Navier-Stokes equations, Phys. Fluids A: Fluid Dyn., 5, 11, 2600-2609, 1993 · Zbl 0809.76078 |
| [82] | Berner, J.; Achatz, U.; Batte, L.; Bengtsson, L.; De La Camara, A.; Christensen, H. M.; Colangeli, M.; Coleman, D. R.; Crommelin, D.; Dolaptchiev, S. I., Stochastic parameterization: Toward a new view of weather and climate models, Bull. Am. Meteorol. Soc., 98, 3, 565-588, 2017 |
| [83] | Branicki, M.; Majda, A. J.; Law, K. J., Accuracy of some approximate Gaussian filters for the Navier-Stokes equation in the presence of model error, Multiscale Model. Simul., 16, 4, 1756-1794, 2018 · Zbl 1408.93132 |
| [84] | Li, Y.; Stechmann, S. N., Predictability of tropical rainfall and waves: Estimates from observational data, Q. J. R. Meteorol. Soc., 146, 729, 1668-1684, 2020 |
| [85] | Harlim, J.; Majda, A., Filtering nonlinear dynamical systems with linear stochastic models, Nonlinearity, 21, 6, 1281, 2008 · Zbl 1147.93411 |
| [86] | Kang, E. L.; Harlim, J., Filtering nonlinear spatio-temporal chaos with autoregressive linear stochastic models, Physica D, 241, 12, 1099-1113, 2012 · Zbl 1300.93168 |
| [87] | Chen, N., Stochastic Methods for Modeling and Predicting Complex Dynamical Systems: Uncertainty Quantification, State Estimation, and Reduced-Order Models, 2023, Springer Nature · Zbl 1515.93004 |
| [88] | Harlim, J.; Majda, A. J., Test models for filtering and prediction of moisture-coupled tropical waves, Q. J. R. Meteorol. Soc., 139, 670, 119-136, 2013 |
| [89] | Chen, N.; Fu, S., Uncertainty quantification of nonlinear Lagrangian data assimilation using linear stochastic forecast models, Physica D, Article 133784 pp., 2023 · Zbl 1515.86013 |
| [90] | Branicki, M.; Chen, N.; Majda, A. J., Non-Gaussian test models for prediction and state estimation with model errors, Chin. Ann. Math. Ser. B, 34, 1, 29-64, 2013 · Zbl 1316.60055 |
| [91] | Chen, N.; Majda, A. J., Model error in filtering random compressible flows utilizing noisy Lagrangian tracers, Mon. Weather Rev., 144, 11, 4037-4061, 2016 |
| [92] | Janjić, T.; Bormann, N.; Bocquet, M.; Carton, J.; Cohn, S. E.; Dance, S. L.; Losa, S. N.; Nichols, N. K.; Potthast, R.; Waller, J. A., On the representation error in data assimilation, Q. J. R. Meteorol. Soc., 144, 713, 1257-1278, 2018 |
| [93] | Uppala, S. M.; Kållberg, P.; Simmons, A. J.; Andrae, U.; Bechtold, V. D.C.; Fiorino, M.; Gibson, J.; Haseler, J.; Hernandez, A.; Kelly, G., The ERA-40 re-analysis, Q. J. R. Meteorol. Soc.: a J. Atmos. Sci., Appl. Meteorol. Phys. Oceanogr., 131, 612, 2961-3012, 2005 |
| [94] | Kalman, R. E., A new approach to linear filtering and prediction problems, J. Basic Eng., 82, 35-45, 1960 |
| [95] | Liptser, R. S.; Shiryaev, A. N., Statistics of Random Processes II: Applications, 2013, Springer Science & Business Media |
| [96] | Chen, N.; Majda, A. J., Conditional Gaussian systems for multiscale nonlinear stochastic systems: Prediction, state estimation and uncertainty quantification, Entropy, 20, 7, 509, 2018 |
| [97] | Chen, N., Learning nonlinear turbulent dynamics from partial observations via analytically solvable conditional statistics, J. Comput. Phys., 418, Article 109635 pp., 2020 · Zbl 07506185 |
| [98] | Petrie, R., Localization in the ensemble Kalman filter, (MSc Atmosphere, Ocean and Climate University of Reading, Vol. 460, 2008) |
| [99] | Houtekamer, P. L.; Mitchell, H. L., Ensemble Kalman filtering, Q. J. R. Meteorol. Soc.: a J. Atmos. Sci., Appl. Meteorol. Phys. Oceanogr., 131, 613, 3269-3289, 2005 |
| [100] | García-Garrido, V. J.; Wiggins, S., Lagrangian descriptors and the action integral of classical mechanics, Physica D, 434, Article 133206 pp., 2022 · Zbl 1498.37125 |
| [101] | Silverman, B. W., Density Estimation for Statistics and Data Analysis, 1986, CRC Press · Zbl 0617.62042 |
| [102] | Vortmeyer-Kley, R.; Holtermann, P.; Feudel, U.; Gräwe, U., Comparing Eulerian and Lagrangian eddy census for a tide-less, semi-enclosed Basin, the Baltic Sea, Ocean Dyn., 69, 701-717, 2019 |
| [103] | Tsagris, M.; Beneki, C.; Hassani, H., On the folded normal distribution, Mathematics, 2, 1, 12-28, 2014 · Zbl 1425.62025 |
| [104] | Schlueter-Kuck, K. L.; Dabiri, J. O., Model parameter estimation using coherent structure colouring, J. Fluid Mech., 861, 886-900, 2019 · Zbl 1415.86033 |
| [105] | Majda, A.; Wang, X., Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, 2006, Cambridge University Press · Zbl 1141.86001 |
| [106] | Branicki, M.; Mancho, A. M.; Wiggins, S., A Lagrangian description of transport associated with a front-eddy interaction: Application to data from the north-western mediterranean sea, Physica D, 240, 3, 282-304, 2011 · Zbl 1217.86005 |
| [107] | Van Emden, H. F., Statistics for Terrified Biologists, 2019, John Wiley & Sons |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
